Download TIME ASYMMETRY IN ELECTRODYNAMICS AND COSMOLOGY

TIME ASYMMETRY IN ELECTRODYNAMICS AND
COSMOLOGY
J. V. NARLIKAR
Institute of Theoretical Astronomy, University of Cambridge,
Cambridge, England
ABSTRACT
The concept of the barrow of time' is discussed in relation to thermodynamics,
electrodynamics and cosmology Time symmetric electrodynamics, the absorber
theory of radiation and quantum transitions receive attention to develop
a working theory applicable to a universe with a perfect future absorber
and an imperfect past absorber, such as the steady state cosmological model.
It is stated that the approach adopted herein establishes a strong connection
between the electrodynamic and cosmological arrows of time, and points
a way towards linking these arrows with the thermodynamic one.
1. THE ARROWS OF TIME IN PHYSICS
What do we mean by the arrow of time? in a somewhat vague manner, we
may associate it with the sense in which we relate our subjective experiences
to the space-time manifold. It would be rash to attempt a precise definition
that will satisfy all scientists and philosophersl it is possible, however, to
discuss the subject in a rigorous manner in the more restricted field of
fundamental physics.
Given a space/time diagram, we can arbitrarily choose a direction along
the time axis, by identifying one end with the past and the other with the
future*. This enables us to arrange events in a chronological order. Consider
a set of events A,B,C,... say, which can occur again and again. If these
events always occur with the same chronological order we are able to use
these events themselves to fix the time sense. This would not be possible
if these events occurred in a random order at different times. Our physical
experience reveals events of both types—although it is the events of the
former type that are responsible for the arrow of time. What are these events?
These events can be broadly grouped under three categories, thermodynamics, electrodynamics and cosmology. As an example of the events in
the first category, consider a hot body and a cold body in contact with each
other. If we observe the system a little later', we find the hot body a little
cooler and the cold body a little hotter than 'before'. In other words. if
we measure the temperatures of the two bodies at different instants we can
order these instants chronologically. Such events are usually known as
*
I am excluding here cosmological models with closed time-like lines.
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J. V. NARLIKAR
irreversible events. In electrodynamics, if we observe an accelerated electric
charge we notice that it loses energy. Thus the energy flow from the charge
can be used to fix a chronological order. In cosmology the expansion of the
universe enables us to do the same. If we photograph two nebulae at instants
t1 and t2, their separation from each other would tell us whether t2 >
or t2 < t1. Thus events in each category determine an arrow of time.
The three arrows described above appear to arise from quite different
sources. We may raise the question: 'Why do these arrows point the way
they do?' In other words, is there any connection between these different
arrows of time? Before we search for a solution of the more fundamental
question: 'Why an arrow of time?' it may well be profitable to investigate
whether the arrows of time in thermodynamics, electrodynamics and
cosmology are mutually related.
Opinions differ as to how deep this connection is. Some physicists believe
that the connection, if any, is very superficial. I want to discuss the opposite
point of view, particularly in connection with electrodynamics and cosmology.
2. TIME SYMMETRIC ELECTRODYNAMICS
Returning to the problem of the radiating charge, we may enquire as to
the origin of the time asymmetry. Maxwell's equations are time symmetric.
Corresponding to any solution of these equations we may obtain another
by changing t to — t. Now the problem of radiating charge is solved by means
of retarded solutions of Maxwell's equations. If instead we use advanced
solutions we would describe a charge receiving energy. Mathematically
both the solutions (and any linear combination of them) describe the electromagnetic fields associated with an accelerated charge. Only the retarded
solution is taken to represent the actual situation. So long as there is no other
direction of time we cannot distinguish between the pure retarded and pure
advanced solutions. Given another arrow of time, however, say from
cosmology, the distinction between the retarded and advanced solutions
becomes real. Why do we choose retarded solutions? The usual answer is
causality'. However, to say that we choose retarded solutions in order to
conform with causality does not take us any further in our understanding
of the electrodynamic arrow of time. So long as the choice of retarded solutions is made on such arbitrary grounds, we will not be able to make progress
in this direction. It appears that to achieve this we have to resort to a different
theory of electrodynamics which does not permit so much freedom of choice.
Such a theory already exists in literature, and in its crude form even
preceded Maxwell's theory. This is the theory of direct interparticle action.
In this theory electric charges interact directly with one another and the
electromagnetic field has no independent existence. Mathematically we may
describe the theory through an action principle due to Fokker1. The action
is given by
—
S=
(1)
(sB) da1 db2
mada
eaeb
Here a, b, c, . .. label the charged particles, ea, ma being the charge and mass
of particle a. A denotes a typical point on the world line of a, da' denoting
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TIME ASYMMETRY IN ELECTRODYNAMICS AND COSMOLOGY
the coordinate displacements and da the proper time element at A, along
the world line. We have used flat space coordinates with metric
= diag. (— 1, — 1, — 1, + 1). The vector indices are raised and lowered
according to 11k• The delta-function in equation 1 has as its argument SB,
the square of the interval between A and B, two typical points on the world
lines of a and b. Thus A and B interact only if they are connected by a light
ray. This is the relativistic generalization of the Newtonian concept of
instantaneous action at-a-distance. The double sum in 1 does not include
the terms a b; the self-action is therefore absent. We have taken the velocity
of light as unity.
The 4-potential generated by charge a at a point X is given by
A(X) = ea$(sj) da
(2)
and it satisfies identically the gauge condition
0
(3)
A = 4J
(4)
—
and the wave equation
where j(X) is the current due to charge a at X
jca)(x) = eac64(A, X)da,
(5)
We may also define the field tensor corresponding to 2
= A}
—
A
(6)
By virtue of relations 3 and 4 this satisfies the Maxwell equations
identically.
This is a fundamental difference between the above theory and the Maxwell
field theory. In the latter, Maxwell equations are genuine equations. Thus
to a solution of equation 7 we can add any solution of
F(a)il ; = 0
(8)
On the other hand in the above theory equation 7 is an identity. The F"
are really given by equations 2 and 6. If we denote the usual retarded solution
generated by charge a to be F, equation 6 corresponds to
F(a) =
[F + FjJ
(9)
Thus there is no freedom of choice; we are forced to accept equation 9.
It was this lack of freedom that proved to be a stumbling block to the
theory for a long time. Charges are observed to produce retarded fields;
not the time symmetric fields given in 9. If we consider the equation of
motion of charge a, obtained by setting S = 0 for a small variation of the
world line of a, we find that a force like the Lorentz force acts on a. But it
arises from the fields of all other particles in the universe, i.e. from a field
a
[F + Fj]
(10)
The advanced fields in equation 10 appear to present an embarrassment to
the theory.
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J. V. NARLIKAR
However, an ingenious way out of this difficulty was found by Wheeler
and Feynman2 with their absorber theory of radiation.
3. THE ABSORBER THEORY OF RADIATION
Wheeler and Feynman argued that the interaction is time symmetric only
between pairs of particles. The possibility still exists that the collective
interaction of a large number of particles could lead to time-asymmetric
results. The actual universe is a collection of large numbers of charged
particles. An asymmetry in the large scale behaviour of the universe is
therefore expected to lead to a local asymmetry in electrodynamics.
According to Wheeler and Feynman the requirement to be met by the
universe in order to produce the necessary asymmetry is that it shall act
as a perfect absorber of all electromagnetic disturbances generated within it.
That is, if we set the charge a in motion, all the disturbances related to this
should be absorbed by the universe:
as r -÷
[F + F] -÷0
faster than 1/r, where r is the distance from a.
Since advanced and retarded fields have supports on different branches
of the light core, expression 11 implies that
—* 0
as r —
— 0,
and hence that
[F — Fj] —* 0
as r - o
However, the quantity in 13 has no sources and hence it satisfies the
homogeneous wave equation. Since it vanishes at infinity faster than 1/r,
it must vanish identically. Therefore the field 10 acting on the particle a
has the form
Jr (b)
ba 2L ret
L
—
advi —
(b)
p(b) J
(a) —
(a)
adv
ba ret 2 L ret
The first term on the RHS denotes the retarded interaction of all other
particles while the second term is the radiative reaction in the form given
I
by Dirac3. Thus the theory appears to provide the correct answer if the
condition 11 of perfect absorption is satisfied.
There was, however, one snag in the argument developed so far. Wheeler
and Feynman applied this to the static Euclidean universe with a uniform
distribution of charges. Such a universe satisfies 11 and hence leads to 14.
However, this universe is time symmetric and hence cannot distinguish
between the past and the future light cones. Hence the time-reversed form of
14, viz.
ba2t + Fj] =
+
—
FJ
is also valid.
Wheeler and Feynman recognized this and argued that the asymmetry
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TIME ASYMMETRY IN ELECTRODYNAMICS AND COSMOLOGY
which allows one to distinguish between 14 and 15 comes from thermodynamics. The thermodynamic arrow of time is ascribed to asymmetrical
initial conditions. It is these conditions which make 14 highly probable
and 15 highly improbable in the sense of statistical mechanics. Thus according
to Wheeler and Feynman the electrodynamic and thermodynamic arrows
of time are strongly related.
A new feature into this argument was introduced by Hogarth4 many
years later. He pointed out that the above argument ignores the cosmological
time arrow. In an expanding universe the past and future light cones do
not behave symmetrically where absorption is concerned. Electromagnetic
waves going into the future are red-shifted while those going into the past
are blue-shifted. Also, in some models of the universe the matter densities
in the past and the future are different. Thus one cannot argue that if 14
holds so would 15 or vice versa. Hogarth's calculations show that in certain
models of the universe, e.g. the open Friedmann models, 14 does not hold
but 15 does. Also in the steady state model 14 holds but not 15. In other
words, the observed alignment of electrodynamic and cosmological time
arrows could be explained only in certain cosmological models, and not
others.
Hogarth's argument had to be improved in two ways, however. First, he
had taken over the Fokker action 1 in flat space and used it in the Robertson—
Walker cosmological spaces of the expanding cosmological models. Since
these spaces are conformally flat, and the electromagnetic equations are
conformally invariant, the above step was justified. For completeness it was
necessary, however, to rewrite equation 1 in the curved space of general
relativity. Secondly, the use of the refractive index, which plays an important
part in the calculation, involved thermodynamics which Hogarth was trying
to avoid. In a later work Hoyle and Narlikar5 took account of both these
points, although their conclusion did not differ from Hogarth's. I summarize
below the main points of the argument.
Suppose we want to arrive at 14. Then the condition to be satisfied by the
universe is that it must be a perfect absorber of all disturbances connected
with the motion of a along the future light cone of a. That is
as r — cc
F —* 0
(16)
faster than 1/r. However, expression 16 requires that we have a definite
process of absorption. If we stick to pure electrodynamics this is provided
by radiative damping. Now the damping term has the correct sign if 14 holds,
and the wrong sign (implying growth of the disturbance) if 15 holds. To avoid
ambiguity therefore it is necessary to assume that 14 holds to begin with and
then to show that the theory is consistent with this assumption. Thus one
has to work within a self-consistent cycle of argument.
When the charge a is set in motion, its retarded field sets other charges
in motion along its future light cone. Since 16 implies a damping of the
retarded field of a as the wave goes away from it, this also implies that the
advanced fields arising from the motion of all other charges b provoked by
the retarded field of a must also be damped. Hence we have
F—0 as r—cc
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(17)
J. V. NARLIKAR
faster than 1/r. Notice that 17 arises from 16 which is the more basic condition. If we were discussing the consistency of 15 we would require 17
to hold along the past light cone of a and then deduce 16. Also, for 17 to
follow from 16 it is essential that the motion of charge a be bounded in
space-time, a requirement not met by the run-away' solutions. Such
solutions are therefore not possible in this theory.
From 16 and 17 we then arrive at 14 through steps similar to those used by
Wheeler and Feynman. It is instructive, however, to see the precise part
played by the universe.
Suppose we are in a universe in which 14 holds. Consider a charge a
at time t = 0. The future light cone of a has t> 0, and the past light cone
t < 0. Now we rewrite the LHS of 14 in the following form
+ FV] = 2ba
ba2
ret + baF
t<O
(18)
t>O
The second term on the RHS of 18 is the advanced contribution of all the
particles b on the future light cone of a, which are excited by the retarded
disturbance from a. This contribution arrives instantaneously as the retarded
wave leaves a. We shall call it the 'response' of the universe and denote it
by R. A comparison with the RHS of 14 shows that
R(a) = 2 b>a F + [F —
t<O
—
—
=
Ii (b)
,'(b)
advi
2 1 ret
ba
t<O
—
F]
Ii
21(a)
ret
adv
Fj]
In 19 we have used the fact that Fj = 0 for t <0.
Equations 19 denotes the response required from the universe in order
to arrive at 14 which is in conformity with observations. As mentioned before
the steady state universe provides the appropriate response, but not the
usual big bang models.
4. QUANTUM CONSIDERATIONS
We have so far considered classical electrodynamics. Can these ideas
be extended to quantum electrodynamics? Recent work6 shows that it is
possible to have a quantum theory of direct interparticle action. The shortage
of space does not permit a detailed discussion of these new ideas. I will be
content with a brief mention of salient features.
The quantum analogue of the radiating electron occurs in the spontaneous
transition of an atomic electron. While the induced transitions are both
upward and downward with symmetrical rates, the spontaneous transitions
are always downward, accompanied by emission of radiation. Can this be
related to the expansion of the universe?
In the Maxwellian quantum electrodynamics the phenomenon of spontaneous transition is explained by first quantizing the electromagnetic field.
The vacuum state in this theory has non-trivial properties as a result of the
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TIME ASYMMETRY IN ELECTRODYNAMICS AND COSMOLOGY
field quantization rules, and these play an important part in arriving at the
spontaneous transition probability. For example, if the electron is making
transitions between two energy levels Em and E, E > Em, in the presence
of a field of n quanta of frequency V = (E — Em)/h the probabilities of
downward and upward transitions are related by
P(EnEm) fl+ 1
P(EmEn)
'
(20
In the direct particle theory there are no fields and hence no field quantization or quanta in the above sense. So far as induced transitions are concerned
the theory gives the same result as field theory. But what about spontaneous
transitions? Returning to the classical expression 14, the first term on the RHS
is responsible for induced transitions whereas the second term leads to
spontaneous transitions. Since the second term is part of the response R
of the universe, it is to the universe that we must look for an explanation of
spontaneous transitions.
It turns out that the calculations are most easily performed by using the
path integral method of Feynman7'8 This provides the quantum analogue
of 14 and 19. For example, the response of the universe is non-zero even
when there is no net electric field from all other particles. Thus the electron
is influenced by the universe and makes a transition (spontaneously).
The physical reason for this may be understood in the following way. A
downward spontaneous transition is accompanied by the emission of an
electromagnetic wave which causes transitions in the future absorber.
An expanding universe acts as a scold environment', with all absorber
particles in their ground states. The wave emitted by the source particle a
is absorbed by the absorber particles which make upward transitions,
induced by this wave. Complete absorption is necessary if the whole cycle
is to work self-consistently. It is also easy to see why spontaneous upward
transitions do not occur. These would require augmentation instead of
absorption of the electromagnetic waves emitted by the source. The universe
in the future light cone does not permit this—since the particles are mostly
in their ground state and cannot jump down farther to enhance the incoming
radiation.
Apart from absorption, the universe also plays an important part through
dispersion. If we Fourier analyse the outgoing wave from the source, the
different components travel at different velocities through the intergalactic
medium. The intergalactic density and other parameters are such that large
phase differences are built up between neighbouring frequencies, with the
result that the different components become randomly phased. This random
phasing is usually assumed in field quantization. In the above work this extra
assumption is not required.
I have so far not mentioned thermodynamics explicitly. But a connection
between the thermodynamic and cosmological arrows now begins to appear.
An expanding universe acts as a sink—a fact which appears to be connected
with the asymmetry of upward and downward transitions considered above.
This asymmetry leads to a result like 20 even in the direct particle theory.
It is well known that the law of black body radiation follows from 20. As
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J. V. NARLIKAR
yet these ideas are somewhat tentative, but they serve to indicate that a
connection between thermodynamics and cosmology may be built up.
5. CONCLUSION
The ideas described above work in a universe with a perfect future absorber
and an imperfect past absorber. Of the well known cosmological models
only the steady state model meets this requirement. If we regard the present
astronomical evidence as unambiguously against the steady state theory
we must abandon this approach altogether. On past record of astronomical
observations, and with the present uncertainties attached to the recent data,
such an unequivocal conclusion is not justified9.
On the other hand we may ask whether there are any gains in the above
approach. As discussed above this approach establishes a strong connection
between the electrodynamic and cosmological arrows of time, and also
points a way towards linking these arrows with the thermodynamic one.
Such a connection cannot be established with Maxwellian electrodynamics
which allows too much freedom of choice. Moreover, as the recent work'°
shows this approach is better than the usual quantum electrodynamics in
handling the so-called radiative corrections.
I end with a few remarks on the question why an arrow of time'. If we
were able to establish a strong connection between the different arrows of
time, physical, biological etc., we could then argue in the following way.
An absolute direction is immaterial: the important concept is the relative
orientation of the different arrows. For example, if we can say 'we grow old
because the universe expands', then this is equivalent to saying that we
grow young as the universe contracts! The question Why do we grow older,
not younger?' without reference to other arrows of time has no meaning.
REFERENCES
2
6
A. D. Fokker, Z.Phys. 58, 386 (1929).
A. D. Fokker, Physica, 9, 83 (1929); 12, 145 (1932).
j• A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945).
P. A. M. Dirac, Proc. Roy. Soc. A, 167, 148 (1938).
J. E. Hogarth, Proc. Roy. Soc. A, 267, 365 (1962).
F. Hoyle and J. V. Narlikar, Proc. Roy. Soc. A, 277, 1(1963).
F. Hoyle and J. V. Narlikar, Ann. Phys. (N. Y.), 54, 207 (1969).
R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals. McGraw-Hill:
New York (1965).
F. Hoyle, Proc. Roy. Soc. A, 308, 1(1968).
10 F.
Hoyle and J. V. Narlikar, Nature, London, 222, 1040 (1969).
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